Simple guide to leveling in FFXIV Heavensward

Level 1 - 8 : Hunt log
Level 8 - 10: Random fates in a starter area.
Level 10 - 13: Hunt Log
Level 13 - 20: Fates in Aleport
Level 20 - 22: Hunt Log (Ask a friend or stranger for help to finish it.)
Level 22 - 28: Leves in Quarrymill, Two in particular. A level 20 one that involves collecting traps and a level 25 one that involves collecting water. Neither require killing so you can turn it up to max level.
Level 28 - 34: Leves in Costa Del Sol. There are three easy Leves that are faster and offer more XP than fates. Out to Sea, which involves click 5 items, no combat. Necrolopis, which you can do with only having to kill one monster and Culling the Herd which involves killing five monsters. The goal is to do Out to sea as much as you can, if its not available, do Necolopis, if that's not available do Culling the herd as a last resort. If Cancer spawns, join a fate group and do that.
Level 34 - 42: Fates in Central Coerthas Highlands
Level 42 - 50: Fates in North Thanalan. I recommend waiting until level 42 before heading here.
Level 44 - 48: DD first room runs if you can create a premade party. Ideal setup: Dark Knight, Astrologian, Summoner, Machinist.
Level 50 - 55: Fates in Western Coerthas Highlands.
Level 53 - 55: Sohm Al
Level 55 - 58: Fates in Churning Mist.
Level 55 - 57: The Aery
Level 58 - 60: Fates in Dravanian Hinterlands
Level 57 - 59: The Vault
Level 59 - 60: Great Gubal Library

 

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Challenge Log: Be sure to unlock and do the challenge log each week. Its essentially a free level or two. The challenges experience reward scale on your level, so its best to use them 30+ in my opinion.

On a side note. I highly recommend using dungeons to level 53+ and doing an occasional dungeon or two every 5 or 10 levels so you have an idea on how to play your class. I can't stand it when I get someone in a party post level 40 who has no idea how to play their class well.

Also, a few tips
The Cancer fate thing is okay if there's a large amount of fate groups, but on small or medium sized servers, this likely isn't the case, and it's better to ignore it.

This same methodology applies to the Big Boss Fates in the Heavensward areas that give rewards for finishing them. Yeah, I know they give achievements, but in the time it took you to kill it, you could've done more fates elsewhere and gotten more exp. If your goal is leveling, then LEVEL. Arguments can be made for some of these fates, but one you ABSOLUTELY should avoid is "Metal Gears Revengeance 2" in Dravanian Hinterlands, as the reward is only slightly better than 2 of any other fate in the area, which you can usually complete more than that by the time MGR2 is finished.

If you feel that 41-42 is too slow using fates in CCH (because it is), then run Stone Vigil. Not only is it good practice for your class, it's fast. With the changes to trash and boss exp with 3.0, NOT killing everything and just going straight through to bosses is very good exp per run. To those that aren't aware, trash exp was reduced and boss exp was increased for all dungeons from Cutter's Cry onwards.

If you're feeling masochistic or have a group you trust, Aurum Vale gives more exp than Dusk Vigil, but the gear isn't as good. This is useful to know for those servers that don't ever seem to have WCH fate groups.